#399221

order	512
nilpotency class	3
isoclinism	isoclinic class of #394108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14777
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,12,7,13,8,14)(3,15,18,32,20,33,19,9)(4,21,23,39,24,10,29,36)(5,25,26,11,28,37,27,38)(16,40,43,34,44,57,49,30)(17,45,46,41,47,31,48,55)(22,51,52,35,53,59,54,50)(42,56,61,64,62,58,63,60);(1,3,8,18,7,20,6,19)(2,9,14,15,13,32,12,33)(4,16,29,43,24,44,23,49)(5,17,26,48,28,47,27,46)(10,34,39,57,21,30,36,40)(11,31,37,41,38,45,25,55)(22,42,52,63,53,62,54,61)(35,58,59,64,50,56,51,60);(1,4,8,29,7,24,6,23)(2,10,14,39,13,21,12,36)(3,16,18,43,20,44,19,49)(5,22,26,52,28,53,27,54)(9,30,15,40,32,34,33,57)(11,35,37,59,38,50,25,51)(17,42,48,63,47,62,46,61)(31,56,41,60,45,58,55,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,26)(9,31)(10,35)(12,25)(13,38)(14,37)(15,41)(16,42)(18,48)(19,46)(20,47)(21,50)(23,54)(24,53)(29,52)(30,56)(32,45)(33,55)(34,58)(36,51)(39,59)(40,60)(43,63)(44,62)(49,61)(57,64)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285493, C₂×#399221, #72285720, …
maximal normal subgroup of	#72285493, C₂×#399221, #72285720, …
maximal quotient of	C₂×#399221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306417);